In mathematics, a holomorphic discrete series representation is a
discrete series representation
In mathematics, a discrete series representation is an irreducible unitary representation of a locally compact topological group ''G'' that is a subrepresentation of the left regular representation of ''G'' on L²(''G''). In the Plancherel meas ...
of a
semisimple Lie group
In mathematics, a Lie algebra is semisimple if it is a direct sum of modules, direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper Lie algebra#Subalgebras.2C ideals and homomorphisms, i ...
that can be represented in a natural way as a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
of holomorphic functions. The simple Lie groups with holomorphic discrete series are those whose symmetric space is
Hermitian {{Short description, none
Numerous things are named after the French mathematician Charles Hermite (1822–1901):
Hermite
* Cubic Hermite spline, a type of third-degree spline
* Gauss–Hermite quadrature, an extension of Gaussian quadrature meth ...
. Holomorphic discrete series representations are the easiest discrete series representations to study because they have highest or lowest weights, which makes their behavior similar to that of finite-dimensional representations of compact Lie groups.
found the first examples of holomorphic discrete series representations, and classified them for all semisimple Lie groups.
and described the characters of holomorphic discrete series representations.
See also
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Quaternionic discrete series representation In mathematics, a quaternionic discrete series representation is a discrete series representation of a semisimple Lie group ''G'' associated with a quaternionic structure on the symmetric space of ''G''. They were introduced by .
Quaternionic disc ...
References
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External links
*{{citation , title=Some facts about discrete series (holomorphic, quaternionic) , url=http://www.math.umn.edu/~garrett/m/v/facts_discrete_series.pdf , first= Paul , last=Garrett , year=2004
Representation theory of Lie groups